Precalculus Examples

$\log_{b} (\frac{p^{4} q^{7}}{m^{3} b^{8}})$

Step 1

Rewrite $\log_{b} (\frac{p^{4} q^{7}}{m^{3} b^{8}})$ as $\log_{b} (p^{4} q^{7}) - \log_{b} (m^{3} b^{8})$ .

$\log_{b} (p^{4} q^{7}) - \log_{b} (m^{3} b^{8})$

Step 2

Rewrite $\log_{b} (p^{4} q^{7})$ as $\log_{b} (p^{4}) + \log_{b} (q^{7})$ .

$\log_{b} (p^{4}) + \log_{b} (q^{7}) - \log_{b} (m^{3} b^{8})$

Step 3

Expand $\log_{b} (p^{4})$ by moving $4$ outside the logarithm.

$4 \log_{b} (p) + \log_{b} (q^{7}) - \log_{b} (m^{3} b^{8})$

Step 4

Expand $\log_{b} (q^{7})$ by moving $7$ outside the logarithm.

$4 \log_{b} (p) + 7 \log_{b} (q) - \log_{b} (m^{3} b^{8})$

Step 5

Rewrite $\log_{b} (m^{3} b^{8})$ as $\log_{b} (m^{3}) + \log_{b} (b^{8})$ .

$4 \log_{b} (p) + 7 \log_{b} (q) - (\log_{b} (m^{3}) + \log_{b} (b^{8}))$

Step 6

Expand $\log_{b} (m^{3})$ by moving $3$ outside the logarithm.

$4 \log_{b} (p) + 7 \log_{b} (q) - (3 \log_{b} (m) + \log_{b} (b^{8}))$

Step 7

Expand $\log_{b} (b^{8})$ by moving $8$ outside the logarithm.

$4 \log_{b} (p) + 7 \log_{b} (q) - (3 \log_{b} (m) + 8 \log_{b} (b))$

Step 8

Logarithm base $b$ of $b$ is $1$ .

$4 \log_{b} (p) + 7 \log_{b} (q) - (3 \log_{b} (m) + 8 \cdot 1)$

Step 9

Multiply $8$ by $1$ .

$4 \log_{b} (p) + 7 \log_{b} (q) - (3 \log_{b} (m) + 8)$

Step 10

Simplify each term.

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Step 10.1

Apply the distributive property.

$4 \log_{b} (p) + 7 \log_{b} (q) - (3 \log_{b} (m)) - 1 \cdot 8$

Step 10.2

Multiply $3$ by $- 1$ .

$4 \log_{b} (p) + 7 \log_{b} (q) - 3 \log_{b} (m) - 1 \cdot 8$

Step 10.3

Multiply $- 1$ by $8$ .

$4 \log_{b} (p) + 7 \log_{b} (q) - 3 \log_{b} (m) - 8$